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Neutron–Antineutron Oscillation and Baryonic Majoron: Low Scale Spontaneous Baryon Violation

Neutron–Antineutron Oscillation and Baryonic Majoron: Low Scale Spontaneous Baryon Violation

Eur. Phys. J. C (2016) 76:705 DOI 10.1140/epjc/s10052-016-4564-0

Regular Article - Theoretical Physics

Neutron– oscillation and baryonic majoron: low scale spontaneous violation

Zurab Berezhiani1,2,a 1 Dipartimento delle Scienze Fisiche e Chimiche, Università dell’Aquila, Via Vetoio, Coppito, 67100 L’Aquila, Italy 2 INFN, Laboratori Nazionali Gran Sasso, Assergi, 67100 L’Aquila, Italy

Received: 30 May 2016 / Accepted: 11 November 2016 © The Author(s) 2016. This article is published with open access at Springerlink.com

Abstract We discuss the possibility that baryon number B n–n˜ oscillation was suggested in Ref. [3], followed by other is spontaneously broken at low scales, of the order of MeV types of models as e.g. in [4–6]. or even smaller, inducing the –antineutron oscilla- Clearly, the existence of the Majorana of the neutron tion at the experimentally accessible level. An associated would violate the conservation of baryon number B by two Goldstone –baryonic majoron can have observable units (analogously, Majorana for violate effects in neutron to antineutron transitions in nuclei or dense number L by two units). If B and L were exactly con- nuclear . By extending baryon number to an anomaly- served, the phenomena like decay, n–n˜ oscillation or free B − L symmetry, the baryo-majoron can be identified neutrinoless double-beta decay would be impossible. Exper- with the ordinary majoron associated with the spontaneous imental limits on matter stability tell that B-violating pro- breaking of lepton number, and it can have interesting impli- cesses must be very slow: lower bounds on the lifetime of the cations for neutrinoless 2β decay with the majoron emission. (and stable nuclei) land between 1030–1034 years We also discuss the hypothesis that baryon number can be [7]. On the other hand, we have a strong theoretical argument spontaneously broken by QCD itself via the six- con- that baryon number must be indeed violated in some pro- densates. cesses. As was shown by Sakharov, B-violating processes which break also CP and which were out of equilibrium at some early cosmological epoch, can generate non-zero 1 Introduction baryon number in the universe [8,9]. Without B-violation no primordial baryon asymmetry could be generated after There is no fundamental principle that can prohibit neutral inflation, the universe would remain baryon symmetric and as are the neutron or neutrinos to have a Majorana thus almost empty of matter. (In modern theoretical scenar- − mass as envisaged long time ago by Majorana [1]. Nowa- ios, B L violation is indispensable and also sufficient [10].) days the neutron is known to be a composite having One cannot exclude that primordial baryogenesis is related − a Dirac nature conserving baryon number. As for the neutri- to the same B and B L violating physics, containing CP- ˜ nos, theorists prefer to consider them as Majorana particles violating interactions of , as also induces n–n mixing ˜ though no direct experimental proofs for this were obtained as e.g. in the models of [3–6].Therelationofn–n physics yet (e.g. the neutrinoless double-beta decay). On the other with P and CP violation was discussed in Ref. [11]. hand, it is not excluded that the neutron n, along the large The structure of the describing the known Dirac mass term m nn, m ≈ 940 MeV,has also a small Majo- particles and their interactions nicely explains why the B T and L violating processes are suppressed. Under the stan- rana mass term  ˜ n Cn + h.c. =  ˜ n n˜ + h.c.,  ˜  m, nn nn nn = ( ) × ( ) × ( ) which mixes the neutron and antineutron states (here C is the dard gauge group G SU 3 SU 2 U 1 ,theleft- charge conjugation matrix and n˜ = CnT is the antineutron handed (LH) quarks and transform as iso-doublets = ( , ) = (ν, ) field). This mixing induces the very interesting phenomenon qL u d L ,lL e L while the right-handed (RH) ones 1 of neutron–antineutron oscillation, n →˜n, suggested a long are iso-singlets u R, dR, eR. As usual, one assigns a global = = / time ago by Kuzmin [2]. The first theoretical scheme for the lepton charge L 1 to leptons and a baryon charge B 1 3

1 For simplicity, in the following we omit the symbols L (left) and R a e-mail: [email protected] (right), the charge conjugation matrix C, and the internal gauge, spinor, 123 705 Page 2 of 10 Eur. Phys. J. C (2016) 76:705 to quarks, so that composed of three valence quarks can be involved, these operators give rise to mixing phe- have a baryon number B = 1. nomena also for other neutral baryons, e.g. oscillation of the However, L and B are not perfect quantum numbers. They into the antihyperon ˜ . are related to accidental global symmetries U(1)L and U(1)B If the scale M9 is taken of the order of the GUT scale, possessed by the standard model Lagrangian at the level of as one takes for the proton decaying operators O6 or for the renormalizable couplings (no renormalizable coupling can be leptonic operator O5 (1), the effects of n–n˜ mixing would written that could violate them). However, they can be explic- become vanishingly small. On the other hand, the GUT scale itly broken by higher dimension (nonrenormalizable) opera- is not really favored by the primordial baryogenesis. The tors suppressed by large mass scales which may be related to latter preferably works at smaller scales, in the post-inflation the scales of new physics beyond the standard model [12]. For epoch. An adequate scale for baryogenesis in the context of example, grand unified theories (GUTs) introduce new inter- some B = 2 models [5,6] can be as small as M9 ∼ 1 PeV. actions that transform quarks into leptons and thus induce Taking into account that the matrix elements of oper- D = O = 1 qqql O effective four-fermion ( 6) operators 6 M2 , ators 9 between the neutron states are of the order of → π + → ν 6 ∼ −4 6 etc. which lead to the proton decays like p e , p K QCD 10 GeV , one can estimate (modulo the Clebsch etc. These decay rates are suppressed by the GUT scale coefficients O(1)): ≥ 15 M 10 GeV, which makes them compatible with the   6 6 5 existing experimental limits [7]. QCD 10 GeV −25  ˜ ∼ ∼ × 10 eV . (4) It is well known that the lowest dimension B − L violating nn M5 M 9 9 operator (D = 5) is related to leptons and it violates the lepton number by two units (L = 2) [12]: The coefficients of matrix elements ˜n|O9|n for different Lorentz and color structures of operators (3) were studied in 1 Ref. [13] but we do not concentrate here on these particu- O5 = lφlφ + h.c. (1) M5 larities and take them as order 1 factors. In the presence of mixing nn˜ (n n˜ +h.c.), the neutron√ mass eigenstates become√ where φ is the Higgs doublet and M is some large mass scale. 5 two Majorana states n+ = (n+˜n)/ 2 and n− = (n−˜n)/ 2, After inserting the Higgs VEV φ=v ∼ 100 GeV, this respectively, with the masses m +nn˜ and m −nn˜ . The char- operator induces small Majorana masses for the neutrinos, acteristic time of n →˜n oscillation in vacuum is related to   τ = −1 2 14 their mass splitting, nn˜ nn˜ . v 10 GeV 8 mν ∼ ∼ × 0.1eV. (2) The direct experimental limit τnn˜ > 0.86 × 10 s (90% M5 M5 C.L.), obtained by a search of n–n˜ oscillation with cold neu- trons freely propagating under the conditions of suppressed The experimental range of the masses, mν ∼ 0.1eV − magnetic field [14], implies  ˜ < 7.7 × 10 24 eV. On the or so, points toward M being close to the GUT scale. nn 5 other hand, there are indirect limits from the nuclear stability: The neutron–antineutron mass mixing (n n˜ + h.c.),vio- n–n˜ mixing inside the nuclei would destabilize the latter [15]. lating baryon number by two units, can be related to the effec- In fact, the operator (3) induces annihilation processes of two tive D = 9 operators involving six quarks. In terms of the nucleons in two or more , NN → π’s, which transform standard model fragments u = u , d = d , and q = (u, d) R R L the nucleus with atomic number A into the nucleus with A−2 these B = 2 operators read with emission of pions with total energy roughly equal to two 1   masses. Interestingly, nuclear stability limits trans- O = uddudd + uddqqd + qqdqqd + h.c. (3) ˜ 9 M5 lated into the free n–n oscillation time are not far more strin- 9 gent than the direct experimental limit of Ref. [14]. In par- 8 ticular, the iron decay limit implies τnn˜ > 1.3 × 10 s (90% where M9 is some large mass scale. These operators can have different convolutions of the Lorentz, color, and weak C.L.) [16], while the oxygen limit is about twice stronger, τ > . × 8 isospin indices which are not specified. (Needless to say, the nn˜ 2 7 10 s (90% C.L.) [17]. This settles the present ˜  < . × −24 combination qq in the second term in (5) must be in a weak upper limit on the n–n mixing mass as nn˜ 2 5 10 eV. 1 αβ Thus, one can conclude that n–n˜ oscillation may test the isosinglet combination, qq =  qαqβ = u d where 2 L L M ∼ α, β = 1, 2 are the weak SU(2) indices, while in the third underlying physics up to scales 9 1 PeV,having in mind term qq can be taken in a weak isotriplet combination as that the experimental sensitivity can be improved by an order  ∼ −25 well.) More generally, having in mind that all quark families of magnitude, down to nn˜ 10 eV.For the present status of n–n˜ oscillation and future projects for its search see e.g. Ref. [18]. Footnote 1 continued and family indices whenever this will not cause ambiguities. Antiparti- One can envisage a situation when baryon number is bro- cles will be termed q˜, u˜, d˜,etc. ken not explicitly but spontaneously. In particular, one can 123 Eur. Phys. J. C (2016) 76:705 Page 3 of 10 705 consider a situation when baryon number is associated with MS, having precisely the same gauge quantum numbers as an exact global symmetry U(1)B, which is spontaneously the right-handed down-quark d(R). Consider the Lagrangian broken by a complex scalar field χ with B = 2. Spontaneous terms breaking of global U(1)B should give rise to a Goldstone † 2 β, which can be coined as the baryonic majoron, or Sud+ Sqq+ S d N + MN N + h.c. (5) baryo-majoron, in analogy to the leptonic majoron associated with the spontaneous breaking of global lepton symmetry where qq in the second term is in a weak isosinglet com- ( ) = 1 αβ = α, β = , U 1 L [19], which is widely exploited in neutrino physics. bination, qq 2 qαqβ uL dL where 1 2 In fact, spontaneous baryon violation in the context of n– are the weak SU(2) indices (we omit the charge conjuga- n˜ oscillation and the physics of the baryonic majoron was tion matrix C and Yukawa constants ∼1). Baryon number previously discussed in Ref. [20], in the context of the model is explicitly violated by the Majorana masses MN . Then, [3]. Spontaneous B-violation was discussed also in Ref. [21], at energies E  MS, MN , operators (3) are induced via in terms of the operator qqql (B = 1). The associated Gold- integrating out the heavy states S and N . Thus, modulo the stone boson was named a bary-, for respect of the elec- Yukawa constants in (5), one obtains troweak anomaly of U(1)B. M5 ∼ 4 . In this paper we discuss the possibility of spontaneous 9 MS MN (6) baryon violation at very low scales, order MeV or even less, in which case the baryo-Majoron can have observable con- From the model point of view, the scale M9 ∼ 1 PeV, acces- sequences. Namely, it would induce nuclear decay via the sible via n–n˜ oscillation, may correspond to a “democratic” Majoron emission, related to transition n →˜n + β in dense choice when MN ∼ MS ∼ 1 PeV. However, it can be nuclear matter. Global baryonic number can be naturally obtained in different situations, namely by taking lighter S 14 extended to anomaly-free B − L. The spontaneous breaking and heavier N (e.g. MS ∼ 10 TeV and MN ∼ 10 GeV), or 7 of the latter must be relevant also for the neutrino Majorana heavier S and lighter N (e.g. MS ∼ 10 GeV and MN ∼ 100 masses.2 In this way, the baryonic and leptonic majorons GeV). become in fact the same particle, just the majoron. In this In the above estimations, the mass scales MN and MN context, we briefly discuss implications for leptonic sector as are in principle independent parameters. In fact, the “heavy e.g. neutrinoless two-beta decay with the majoron emission neutrino” N and “heavy neutron” N cannot be the same par- and astrophysical implications of the neutrino decay. At the ticle. Otherwise its exchange would induce also operators end, we also discuss a rather unusual possibility when baryon like uddν with low cutoff scale which would induce unac- number is broken by six-quark condensates uddudd and ceptably fast proton decay. If N and N are assumed to be its possible implications. gauge singlets, then they can be divided by some discrete symmetries. This can be e.g. a Z2 symmetry changing the sign of leptons, l →−l, e →−e, and N →−N, while 2 Seesaw for neutron–antineutron mixing all other fields remain invariant. Equally, one can consider a baryonic Z2 symmetry q, u, d →−q, u, d and N →−N . The contact (nonrenormalizable) L and B violating terms (1) Notice that these symmetries forbid also dangerous Yukawa and (3) can be induced in the context of UV-complete renor- couplings S†ql, which together with the couplings Sud and malizable theories after decoupling of some heavy particles. Sqq would induce too fast proton decay. In particular, the leptonic operator (1) can be induced in the Alternatively, one can consider N as a weak isotriplet and context of by introducing the Yukawa N as a color octet, in which case no mixed mass terms may couplings φNl + h.c. which involve the heavy gauge singlet exist between N and N states. For N being a color octet the N(R), the so-called right-handed neutrinos, with scalars S can be taken as color anti-sextets, in which case 2 † large Majorana mass terms, MN N + h.c., explicitly vio- also the problematic couplings S ql will be automatically N lating L by two units. Then at lower energies E  MN ,the eliminated. The exchange via color octet would generate operator (1) emerges after integrating out the heavy neutrinos operators (udd)8(udd)8 with (udd)8 in a color-octet combi- N and, modulo the Yukawa constants, one gets M5 ∼ MN . nation. Via a Fierz transformation, exchanging d states from One can also discuss a simple seesaw-like scenario for the left and right brackets in this operator, we see that its the generation of B = 2terms(3). Let us introduce a matrix element will contribute to n–n˜ mixing. gauge singlet Weyl fermion (or fermions) N(R), a sort of In the context of , such operators can easily heavy “RH neutron”, and a color-triplet scalar S, with mass be obtained via R-parity breaking terms u AdBdC (B = C) in the superpotential, where A, B, C are the family indices. 2 One could consider global B − L as a limit of a local gauge symmetry Taking e.g. a superpotential term uds involving the up, down, U(1)B−L when its gauge coupling constant is vanishingly small [22]. and strange RH supermultiplets, one obtains the couplings 123 705 Page 4 of 10 Eur. Phys. J. C (2016) 76:705   analogous to Sud + S†dN of (5) with S being the strange v2 1014 GeV mν ∼ ∼ × 0.1eV. (10) squark and N being a with a Majorana mass MN .In MN Vχ fact, the gluino may have flavor-changing coupling between quark and squark states, namely between d-quark and s- Since the neutrino masses favor the scale Vχ ∼ 1014 GeV, −25 squark. Needless to say, in this scheme somewhat bigger according to the estimation (8) one can obtain nn˜ > 10 mixing mass would be generated for , between eV, potentially accessible in the search of n–n˜ oscillation, if ˜ and , via flavor diagonal gluino coupling between s-quark the color scalars S have masses in the range MS ∼ 10 TeV; ˜ and s-squark. However, – mixing is much more difficult this is within the experimental reach for the new accelerators to experimentally detect (though it maybe more efficient in and perhaps also for the LHC.3 the dense nuclear matter in the neutron stars where hyperons Here the following remark is in order. Although the direct ˜ can emerge as natural occupants). In any case, – mixing limits on the heavy color scalars tolerate MS ∼ 1 TeV, their 2 would also induce nuclear instability via two nucleon anni- exchange induces the effective operators (gqq/MS) (qq¯ qq¯ ) + → + hilation processes with emission, N N K K involving light quarks q = u, d, where gqq is the Yukawa etc. constant of the coupling Sqq (or Sud)in(7). These contact Let us consider now a situation when baryon number is terms are conventionally parameterized as (2π/ 2)(qq¯ qq¯ ). broken not explicitly but spontaneously. Namely, let us mod- The current limits on the compositeness scale are already ify the seesaw Lagrangian (5) as follows: above 10 TeV; namely, the recent LHC limits [24] yield >12 TeV or >17.5 TeV depending on their sign, Sud+ Sqq+ S†d N + χN 2 + h.c., (7) i.e. whether these operators have destructive or constructive interference with the QCD processes. Hence, these bounds / > . / > . where we prescribe B =−2/3 to scalars S and B =−1 translate into the limits MS gqq 4 8 TeV or MS gqq 7 0 to extra fermions N , and introduce a complex scalar field TeV. χ with B = 2. In this way, all couplings in (7) respect the Constraints from the flavor violation can be generically stronger but they are rather model dependent. The couplings exact global symmetry U(1)B. The non-zero VEV χ=Vχ spontaneously breaks the baryon number and induces the of scalar S with light quarks, Sud, will not induce flavor violating transitions as sd¯ → ds¯ leading to K –K¯ mix- Majorana mass MN ∼ Vχ . Hence, the operator O9 emerges after the spontaneous baryon violation as shown on Fig. 1, ing etc. However, if the couplings involving heavier quarks with n–n˜ mixing parameter  inversely proportional to the are also present, with constants of order one, then the lat- baryon symmetry breaking scale Vχ : ter transition can be induced by box diagrams which could bring the limits on MS above 100 TeV. However, one can   6 30 5 envisage some clever possibilities for suppressing these tran- QCD 10 GeV −25  ˜ ∼ ∼ × . sitions without fine tunings of the Yukawa constants. Let nn 4 4 10 eV (8) MS MN MS Vχ us give one simple example, making use of chiral hori- zontal symmetry SU(3)L × SU(3)R under which the LH The scale Vχ can be related also to the breaking of lepton quarks q = (u, d)L of three generations transform as a ( ) ( ) number if one extends global symmetry U 1 B to U 1 B−L triplet of SU(3)L , the RH down quarks dR transform as a and considers the following seesaw Lagrangian: triplet of SU(3)R, while the RH up quarks u R as well as the extra fermions N transform as anti-triplets of SU(3)R.The φNl + χ † N 2 + h.c. (9) Yukawa couplings for the fermion masses can be induced

u φ d φ˜ by the operators M u RqL and M d RqL introducing the ( ) × ( ) U( ) − flavon fields in mixed representations of SU 3 L SU 3 R, In this way, after 1 B L breaking we obtain ¯ ¯ ¯ u ∼ (3, 3) and d ∼ (3, 3) having large VEVs with hierar-

3 The masses of N and N states are induced by the same scalar χ and thus they are ∼ Vχ . However, for guaranteeing proton stability, these states should be distinguished by other quantum numbers, as e.g. the leptonic and/or baryonic Z2 symmetries discussed above. Let us also mention an interesting link between seesaw mechanisms for the generation of the neutrino and neutron Majorana masses: in parallel to the usual leptogenesis scenario [23] due to the heavy neutrino decays N → lφ producing lepton number which then induces baryon number via B−L conserving sphaleron effects [10]. Also baryogenesis can take place via the CP-violating effects in N decays mediated via colored Fig. 1 Diagram generating n–n˜ mixing via exchange of N state which scalar S, N → dS¯ and S →¯ud¯, which can directly produce the baryon gets a large Majorana mass MN ∼χ after U(1)B symmetry breaking number of the universe. 123 Eur. Phys. J. C (2016) 76:705 Page 5 of 10 705 chical and misaligned structures, in line with the mechanisms discussed in Refs. [25–28]. As for the states N , they can ( † ) u d AB N AN B get the Majorana masses via the operator M , A, B = 1, 2, 3 being the indices of SU(3)R. In this case the A couplings Sqq are forbidden while Sud = Su dA are diago- nal in the flavor basis. As a result, the box diagrams induced by exchange of the S scalar will have GIM suppression in the mass eigenstate basis, which will make MS ∼ 10 TeV perfectly compatible with the flavor violation limits. Spontaneous breaking of global U(1)B and U(1)L gives rise to Goldstone , baryo-majoron or lepto-majoron.4 These two can be the same particle, simply a majoron, once the global symmetry is promoted to U(1)B−L . However, in practice a very large scale of symmetry breaking renders such majoron(s) unobservable experimentally and without any important astrophysical consequences. In the following section we discuss models where the global symmetry break- ing scale can be rather small, < 1 MeV or less, in which case the majoron interactions with the neutron as well as with neu- Fig. 2 Upper diagram generates n–n˜ mixing in a low scale model via trinos could have observable experimental and astrophysical exchange of the heavy Dirac fermion with mass term M = MD,with consequences. insertion of the VEV χ. In the presence of mirror sector containing the twin quarks u , d connected to N , lower diagram generates n–n mixing via only the Dirac mass term, without insertion of χ field

3 Low scale seesaw model χ †   O ∼ + + + . . 10 2 4 uddudd uddqqd qqdqqd h c Is it possible to build a consistent model in which baryon MD MS number, or B − L, spontaneously breaks at rather low scales, (12) in which case the majoron couplings to the neutrinos and to the neutron can be accessible for the laboratory search? Thus, at low energies these operators reduce to the neutron This can be obtained by a simple modification of the above Yukawa couplings with the scalar χ, considered model. Let us modify the Lagrangian terms (7) by introducing, † T Ynχ n Cn + h.c. (13) along with the Weyl fermion N with B =−1, also another N = Weyl spinor with B 1. These two together can form with the coupling constant a heavy Dirac particle with a large mass M = MD.The     2 2 relevant Lagrangian terms now read 3 13 3 ∼ QCD ∼ 10 GeV × −30. Yn 2 2 10 (14) † 2 † 2 MD MS MD MS Sud + Sqq + S d N + MDNN + χN + χ N + h.c. (11) Once U(1)B is broken by the VEV χ=Vχ , the neutron– antineutron mixing emerges as Both N and N are coupled to the scalar χ (with B = 2)     and get the Majorana mass terms from the VEV of the latter, 26 6 χ= 10 GeV Vχ −24 Vχ . We assume Vχ to be much less than M and MS.  ˜ = χ∼ × . nn Yn 2 4 10 eV In this way, the diagram shown in Fig. 2, after integrating MD MS 1MeV out the heavy fermions, induces D = 10 operators invariant (15) under U(1)B: Hence, if we take MS ∼ 10 TeV and MD ∼ 100 TeV, or 4 −24 In reality, in the frames of the standard model both of these sym- MS ∼ 100 TeV and MD ∼ 1 TeV, then nn˜ ∼ 10 eV metries have electroweak anomalies and thus these majorons will get would require Vχ ∼ 1 MeV or so. ( ) tiny masses of non-perturbative origin. However, U 1 B−L is free of Low scale (but explicit) baryon number violation was sug- anomalies and the corresponding Goldstone particles would exactly keep their masses. In any case, the tiny masses of the majoron will have gested in Ref. [5], in a model which was mainly designed for no observable physical or astrophysical implications. inducing neutron–mirror neutron oscillation n–n where the 123 705 Page 6 of 10 Eur. Phys. J. C (2016) 76:705

mirror neutron n is the mass degenerate twin of the neutron In the limit μ → 0, η gets a VEV η=Vη = m2/λ while which belongs to hypothetical mirror world, a parallel gauge χ is vanishing. However, for μ = 0, the non-zero VEV χ ≈ μ 2/ 2  sector with a particle content identical to that of the ordinary of is also induced,√ Vχ Vη Mχ Vη. In this case particles which is related to ordinary sector via parity trans- we have fβ ≈ Vη/ 2 Vχ . For example, taking m ∼ 1 formation (for reviews, see e.g. [29–31]). Our model in fact MeV, μ ∼ 1 GeV and Mχ ∼ 10 GeV, we get Vχ ∼ 10 generalizes the mechanism of Ref. [5] for the case of sponta- eV against fβ ∼ 1 MeV. Then, in view of Eqs. (14) and neous U(1)B violation. In this case N and N states should (15), for achieving the experimentally testable range for n–n˜ −25 −24 be treated symmetrically: their Yukawa coupling constants mixing, nn˜ ∼ 10 eV or so, we must have Yn ∼ 10 . χ χ † with and in (11) should be equal, while in addition For MS ∼ 10 TeV, this would require MD ∼ 1 TeV or so. the terms should be included that couple N to u , d , and In any case, the majoron β is coupled non-diagonally , S states, twins of u d, and S from the mirror sector. In this between the n and n˜ states, ignβ nγ n˜+h.c., with the Yukawa 5 way, the lower diagram of Fig. 2 induces n–n mixing with constant gn = Yn Vχ /fβ . As far as nn˜ = Yn Vχ , we thus have   the relations5 6 21 5 QCD 10 GeV −16  ∼ ∼ × 10 eV, (16)  ˜ fβ nn 4 4 = nn , = . MD MS MD MS gn Yn gn (19) fβ Vχ which corresponds to an n–n oscillation time of τ ∼ 10 s. nn Therefore, for fβ Vχ , the Yukawa coupling (13)ofthe Hence, n–n mixing, which conserves a combined baryon scalar χ can be large, Yn gn, while χ itself can be rather number B¯ = B − B between ordinary and mirror sectors, heavy, Mχ 1 GeV. For example, in the context of the can be a dominant effect, while n–n˜ mixing which breaks B¯ Lagrangian (18)wehaveMχ μ/Vχ fβ . / is suppressed by the ratio Vχ MD: In vacuum the transition n →˜n +β is suppressed as far as the masses of n and n˜ are equal. However, in nuclei the neu- Vχ nn˜ = nn . (17) tron and antineutron have different effective potentials and MD thus a n →˜n transition with majoron emission becomes pos- sible. (This phenomenon is rather similar to matter induced Asamatteroffact,n–n mixing can indeed be much larger neutrino decay with majoron emission [44,45].) The decay than n–n˜. Existing experimental limits on the n–n transition rate of the neutron in nuclei can be estimated as [32–36] allow the neutron–mirror neutron oscillation time to be less than the neutron lifetime, with interesting implications 2 gn for astrophysics and particle phenomenology [5,37–43]. (n →˜nβ) = Enn˜ , (20) 2π Let us discuss the physics of the baryo-majoron, β, Gold- stone boson related to the spontaneous breaking of global where Enn˜ =|Un˜ − Un| is a typical energy budget for U(1)B symmetry. Its coupling to emerges by the n →˜n transition (the average difference between the neu- substitution χ = Vχ exp(iβ/fB) in (13). It is worth to notice tron and antineutron potentials inside the nucleus) which that in general the VEV χ=Vχ and the majoron decay is typically of order 10 MeV for lighter nuclei and it can constant fβ are independent parameters. Namely, if U(1)B is reach several 100 MeV for heavier nuclei. (Needless to say, broken solely by the VEV of χ which emerges√ via a negative if the massive component ρ of the scalar χ is light, with mass2 term in its potential, then we get fβ = 2Vχ .Inthis the mass order MeV, then the nuclear transition n →˜n + ρ √ρ casewehaveχ = (Vχ + ) exp(iβ/fB ), where ρ is the mas- is also allowed.) The produced antineutron promptly anni- 2 sive (Higgs) mode with a mass ∼ Vχ , which can be light and hilates with other spectator nucleons producing pions, with which will interact with the same Yukawa constants as the total energy roughly equal to two nucleon masses. majoron. But in the generic case, when besides the scalar χ However, experimentally it is difficult to distinguish n → ˜ +β there are also some other scalars ηi , with baryon charges Bi , n decay from the nuclear decay due to annihilation of two η = which get non-zero√ VEVs i Vi , for the majoron decay 5 The following remark is in order. In general, the operators (12)do β = 2 + ( )2 > χ constant we have 2 f 4Vχ i Bi Vi 2V .For not respect parity and also violate CP. Therefore, after taking the matrix † T example, one can imagine a situation when χ itself has a element n|O10|˜n, in addition the coupling Ynχ n Cn+h.c. (13)there χ † T γ 5 + . . large positive mass2 term, but its VEV is induced by non- can emerge a coupling Xn n C n h c , with generically complex χ  T γ 5 η = constant Xn.AftertheVEV is inserted, the mass term n C n zero VEV Vη of a scalar with B 1, via the coupling term  T † 2 can be rotated away and so only the term nn˜ n Cn is relevant for n– χ η in the Higgs potential: ˜  = 2 + 2 n mass mixing [11], with nn˜ Vχ Yn Xn.However,theterm χ † T γ 5 h λ Yn n C n will contribute the majoron couplings which will have 2 † † 2 2 † † 2 2 2 V(χ, η) = Mχ χ χ + (χ χ) − m η η + (η η) a generic form ignβ n(a + bγ5)n˜, |a| +|b| = 1. Unfortunately, the 2 2 effects of P and CP violation in the majoron couplings can hardly be +μ(χ†η2 + h.c.). (18) detectable in experiments. 123 Eur. Phys. J. C (2016) 76:705 Page 7 of 10 705 nucleons in two or more pions induced by n–n˜ oscillation, since the majoron takes only a small fraction of the energy.6 In the latter case, the nuclear disappearance rate per neutron can be estimated as [15]

2  nn˜ n˜  ˜ (21) nn  2 + ( / )2 Enn˜ n˜ 2  ∼ where n 300 MeV is the antineutron annihilation rate at Fig. 3 Diagram generating the neutrino majorana masses in a low scale nuclear densities. Let us remark also that, for heavier nuclei, majoron model with very large Enn˜ ,therate(21) decreases, while the rate of majoron induced decay (20) is proportional to Enn˜ and so it becomes more effective for heavier nuclei. grating out of the heavy states, one obtains the operator ˜ The present limits on the n–n transition lifetime of the χ 32 neutron bound in nuclei are at the level of 10 years. Namely, O6 ∼ lφlφ + h.c., (23) − 2  1 < . × 31 MD for iron one has nn˜ 7 2 10 (90 % C.L.) [16] and for − oxygen  1 < 1.9 × 1032 (90% C.L.) [17]. For n–n˜ mixing, nn˜ which at lower energies results in the neutrino Yukawa cou-  < . × −24 these limits, respectively, yield nn˜ 1 2 10 eV and χ −24 plings with the scalar , nn˜ < 2.5×10 eV.As far as n →˜n+β decay is regarded, in view of Eq. (20) these limits translate into a bound of χνT ν + . . −31 Yν C h c (24) gn < 10 or so. Imagine now that in future experiments with large volume with detectors the signal for n →˜n is found at the level of a transi- 32 33   tion lifetime of 10 –10 years. This could be interpreted via v2 2 − 100 TeV −6 ˜  ∼ 24 →˜+β ν ∼ ∼ × n–n mixing with nn˜ 10 eV,or via the decay n n Y 2 10 (25) −31 M MD with gn ∼ 10 . However, the first possibility can be dis- D criminated by the direct search of n–n˜ oscillation with free Then for the neutrino Majorana masses we have neutrons at the European Spallation Source (ESS) in which  ∼ −25     a sensitivity down to nn˜ 10 eV can be achieved [18]. 2 100 TeV Vχ Hence, if the result of the direct search at the ESS will be mν = Yνχ∼ × 1eV. (26) negative, then the positive result in nuclei can be interpreted MD 1MeV via the majoron induced decay. Taking into account also uncertainties in the Yukawa con- Now, if one promotes U(1) symmetry to U(1) − in B B L stants in (22), this estimate falls in the experimental mass the context of a low scale model, fβ ≤ 1 MeV, the baryo- range of neutrinos when MD ∼ 100 TeV and Vχ < 1MeV. majoron should be the same particle as the usual (leptonic) − For M ∼ 1 TeV, when Yν ∼ 10 2, one needs Vχ ∼ 10 majoron. The Majorana masses of the neutrinos can be D eV. As we have shown above, this situation can be obtained induced, as in the model suggested in Ref. [48], from the when Vχ is induced by scalars η via the Lagrangian (18) diagram shown in Fig. 3 via the following Lagrangian terms: √ while fβ = 2Vη ∼ 1MeV.8

φ lN + MNN + χ N 2 + χ † N 2 + h.c. (22) Footnote 7 continued where N, N are the fermion couples, analogous to N , N , N, N and “heavy neutrons” N , N must be different particles distin- − guished by some discrete symmetry as e.g. Z2: otherwise their exchange with properly assigned lepton number (or better B L) and would induce the operators like uddν with a low cutoff scale leading 7 they have large Dirac masses M ∼ MD. Then after inte- to dramatically fast proton decay. 8 The following remark is in order. Even if the scalar χ itself has a large 6 Two nucleons can also directly annihilate into the majoron and a , mass Mχ and its small VEV is induced by η, the annihilation process of → β +π β NN , phenomenon resembling the neutrinoless 2 decay with two neutrons in two antineutrinos can take place in the nuclei, with the the majoron emission [46–48], though violating not lepton but baryon ( →¯νν)¯ =σv ∼ 2 2( / )2 × −6 rate nn nnucl Yn Yν 10 GeV Mχ 10 GeV, number. Since the majoron itself is not detectable, a characteristic signal 38 −3 where nnucl ≈ 1.2 × 10 cm is the nuclear density. Confronting this → − − in this case can be provided by the nuclear transition A A 2with with the experimental bounds on dinucleon decay,  1(nn →¯νν)¯ > a single pion emission with energy E ≈ m , while the majoron takes 30 −28 n 10 years [7], we obtain an upper limit YnYν (10 GeV/Mχ )<10 away the same energy. However, the branching ratio of this process is orso.InviewofEqs.(14)and(25), this translates into the bound small with respect to multi-pion annihilation. 1/4 7 2 MD MS(Mχ /10 GeV) > 10 GeV . The Lagrangian model (18) 7 Forsimplicity,wetaketheirDiracmasstermsM and MD parametri- with fβ ∼ 1 MeV and Vχ ∼ 10 eV corresponds to this limit for the cally of the same order. However, we recall that the “heavy neutrinos” benchmark values Mχ ∼ 10 GeV, MS ∼ 10 TeV, and MD ∼ 1TeV. 123 705 Page 8 of 10 Eur. Phys. J. C (2016) 76:705

In this situation, the majoron β can have pretty large Concluding this section, we have shown that in our model Yukawa couplings with the neutrinos, gν = mν/fβ [48]. with low scale B − L breaking, fβ ≤ 1 MeV, the majoron β − Hence, for fβ < 100 keV one could have gν > 10 6 or so, can have large enough Yukawa coupling to neutrons, in the −31 in the range of interest for searching the neutrinoless two- range gn ∼ 10 , as well as to neutrinos, in the range gν > beta decay with the majoron emission [46,47]. The present 10−7 or so. Therefore, such a scenario can be tested in large experimental bound on the majoron coupling to νe reads volume detectors searching for baryon violation via nuclear <(. − . ) × −5 gνe 0 8 1 6 10 [49]. Analogous couplings with instabilities well as in the experiments testing the lepton num- other neutrino flavors can bring about observable effects with ber violation with the neutrinoless two-beta decays, etc. interesting applications for astrophysics and cosmology as However, there arises a naturalness issue questioning how e.g. matter induced neutrino decay [44,45,50–52] or matter B − L can be broken by the VEVs of elementary scalars induced decay of the majoron in two neutrinos [53], block- ≤ 1 MeV, at least five orders of magnitude less than the ing of active–sterile oscillations in the early universe by the electroweak scale MZ ∼ 100 GeV, without invoking fine majoron field [54,55], etc. A detailed analysis of the astro- tuning. Namely, in the absence of other fields like η, when physical limits on the neutrino–majoron couplings can be scalar χ gets a VEV from its own Higgs potential, it should 2 2 found in [56]. have a negative Mχ of the order of 1 MeV . In the context of A few words about cosmological limits. If the majoron the potential V(χ, η (18) when the VEV of χ is induced by coupling constants with neutrinos are large, majorons will the VEV of η, the latter should have negative m2 ∼ 1MeV2. come into equilibrium with neutrinos by reactions νν → ββ The way out is to relate fβ with some compositeness scale. 2 † etc. in the early universe before the neutrino decoupling (T ∼ Imagine that the scalar χ, having positive mass term Mχ χ χ ¯ 2 MeV) and their presence would contribute to the effective with Mχ > 1 GeV, has the Yukawa couplings χ QQwith the number of neutrinos as Nν = 4/7, at the margin of current quark-like states of some hidden sector with a confinement ∼ ≤ ¯ BBN bounds. For the low scale majoron with fβ Vχ scale B ∼ MeV. Then the condensate QQ induces the ¯ 2 1 MeV, this value will be doubled by the contribution of the non-zero VEV χ∼QQ/Mχ , while for the majoron ρ Higgs mode , in evident contradiction with the BBN limits. scalewehave fβ ∼ B. Let us also notice that due to the  = σ Confronting the majoron production rate β nν chiral anomaly of U(1)B−L current with respect to hidden 1/2 2 with the Hubble parameter H = 1.66g∗ T /MPl, where gauge sector, the majoron β will get a mass ∼ B, becoming g∗ = 10.75 is the effective number of particle degrees of a sort of axion for the hidden sector related to its η state. freedom at the temperatures T of few MeV, nν = 0.18T 3 However, the mass of order MeV of β cannot suppress n → is the equilibrium density for one neutrino species, and n˜ + β transitions in nuclei. −3 4 2 σ = 4 × 10 · gν /T is the νν → ββ reaction cross- section, with the Hubble rate H at T > 2 MeV, one obtains − the condition gν = mν/fβ < 3 × 10 5.Thus,inviewof 4 Discussion and outlook the cosmological limits on neutrino masses, mν < 0.3eV, one can see that majorons do not come into equilibrium if At this point, I am tempted to discuss a less orthodox idea, fβ > 10 keV or so. suggesting that the baryon number could be violated in the Certainly, the scalar χ should come into equilibrium also standard model itself, namely by the strong dynamics of the in very early universe, T > M , by interactions with the D QCD sector. The conjecture is that the QCD itself could break heavy fermions N, N etc. with large Yukawa couplings baryon number by two units, by forming a six-quark conden- χ NN etc. in (22). However, it decouples at T < M , while D sate uddudd=B = 9 , along with the basic quark and neutrinos remain in equilibrium with all standard particle B condensates, qq and G2 or higher order operators degrees of freedom with g∗(T < M ) = 110 or perhaps D qGq, qqqq which conserve baryon number. These six- even more if there are new particles with masses less than quark condensates can be built upon different combinations M as e.g. color scalars S. Therefore, at the BBN tempera- D of left and right u, d, and perhaps s quarks, and they may tures, T ∼ 1 MeV, the majorons will have a lower temper- have different convolutions of the Lorenz and color indices. ature than the neutrinos, Tβ /Tν = g∗(T = M )/g∗(T = D Dynamically, they might be induced by the attractive ) 1/3 >( / . )1/3 . 1MeV 110 10 75 2 2 so that two 0 between the electrically neutral three-quark trilinears (udd) β ρ  = . 8 states and together will count as Nν 0 05 extra neu- in color-octet combinations, or as electrically neutral bound χ > trinos. The same applies to the case when is heavy, Mχ state of diquarks in color triplet combinations.9 few GeV,but it Yukawa couplings to neutrinos YνχνT Cν are − large, say Yν ∼ 10 2. In this case the majoron will decouple at the temperatures T ∼ Mχ . Therefore, the BBN limits can 9 With three quark flavors u, d, s, the condensate could appear in flavor be fully respected. singlet combination udsuds as a of the three diquarks ud, us,andds. This would induce the Majorana mass term for the hyperon. 123 Eur. Phys. J. C (2016) 76:705 Page 9 of 10 705

2 < −20 B = 9 = is suppressed by a factor 10 , so that B 2 9 < × ( )9 C QCD C 1MeV . The baryo-majoron β should emerge, as a compos- ite Goldstone mode of this condensate, uddudd= B exp(iβ/fβ ), with fβ ∼ B, exactly like the pions emerge as the Goldstone modes of the quark condensate breaking the chiral SU(2)L × SU(2)R symmetry, qq=  ( τ π / )  ∼ 3 ∼ exp i a a fπ , with QCD and fπ QCD.The majoron coupling constant between n and n˜ states is related to nn˜ via a Goldberger–Treimann-like relation, gn = nn˜ /fβ . Fig. 4 Diagram generating the n–n˜ mixing via baryon-violating six- Therefore, for fβ < 1 MeV, say with 200 keV, the quark condensate uddudd B nuclear stability limits concerning both the values of the mix- ing mass nn˜ and the Yukawa coupling gn can be respected. A non-zero condensate uddudd would induce the An interesting feature of the dynamical baryon violation neutron–antineutron mixing, as shown on Fig. 4. One can by the QCD can be that the order parameter B could be dif- 8 roughly estimate the mixing mass as nn˜ ∼ B/(1GeV) , ferent in vacuum and in dense nuclear matter, i.e. in nuclei or simply taking scales of the neutron mass and residue and all in the interiors of neutron stars. In particular, in dense nuclear relevant momenta order 1 GeV and neglecting combinatorial matter spontaneous baryon violation could occur even if it numerical factors. Hence, for compatibility with the experi- does not take place in vacuum. Or right the opposite, dense −24 mental limit, nn˜ < 2.5 × 10 eV, this condensate must be nuclear matter could suppress the baryon-violating conden- very fuzzy, with a mass parameter B < 1 MeV or so. On sates. In this case, the search of neutron–antineutron oscilla- the other hand, it is believed that any condensate in QCD, if tion with free neutrons and nuclear decay due to the neutron– it appears, must be characterized by the scale QCD ∼ 100 antineutron transition becomes a separate issue. Namely, ¯ ∼ 3 MeV,as e.g. one has for the quark condensate qq QCD. it might be possible that the baryon-violating condensates Thus, we again encounter the problem of hierarchy, at least of evaporate at nuclear densities and do not lead to nuclear insta- 9 9 20 orders of magnitude, between the values B and QCD. bilities, while for free neutrons propagating in the vacuum Formally, the theorem of Vafa and Witten [57] excludes they might be operational. the possibility of baryon number violating condensates in QCD. However, this theorem is based on assumptions which Acknowledgements The idea of this work emerged as a result of leave some loopholes. Namely, the proof of Ref. [57]isfor- numerous discussions with Yuri Kamyshkov. I thank Yuri for motivat- ing me to write it down and for a help in preparation of the manuscript. mally valid if all quarks are massive (in fact, one believes I would like to thank Gia Dvali, Oleg Kancheli, Arkady Vainshtein, and that all light quarks u, d, s have masses of few MeV), and, Andrea Addazi for many valuable discussions. The work was supported remarkably, if at the same time the vacuum angle  is exactly in part by the MIUR triennal grant for the Research Projects of National zero. Interest PRIN 2012CPPYP7 “Astroparticle Physics”. Therefore, one can envisage that in some imaginable world Open Access This article is distributed under the terms of the Creative where the QCD vacuum angle is large,  ∼ 1, a baryon- Commons Attribution 4.0 International License (http://creativecomm   B ∼ 9 ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, violating condensate uddudd could exist, with QCD.  and reproduction in any medium, provided you give appropriate credit In the absence of the axion mode which would relax to zero, to the original author(s) and the source, provide a link to the Creative it could be formed as a dynamical reaction of the system tend- Commons license, and indicate if changes were made. ∼ 2  4 3 ing to decrease the vacuum energy cos QCD associated Funded by SCOAP . to non-zero . Thus, one can envisage that it value depends B = () 9 on the vacuum angle,  F QCD. 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